Tensor products of any number of operators acting on their respective Hilbert spaces can be defined in this manner to an operator that acts on the system described by the tensor product of the spaces. For a register of \(n\) single qubits, the operator on \(\mathcal{H}^{2^n}\) that applies a single qubit operator \(U\) on \(\mathcal{H}^2\) on say the \(i^{th}\) qubit of the register can be written as the \(n\)-fold tensor product
\[I\otimes ...\otimes I\otimes U\otimes I \otimes ... \otimes I,\]
where \(I\) is the identity operator on a single qubit, and \(U\) appears in the \(i^{th}\) position of the tensor product. In this way, the tensor product of single qubit gates can be built up from the sum of matrices of this form.